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In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields, a fundamental step is a factorization of a polynomial over a finite field.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
Square brackets are also often used in place of a second set of parentheses when they are nested—so as to provide a visual distinction. In mathematical expressions in general, parentheses are also used to indicate grouping (i.e., which parts belong together) when edible to avoid ambiguities and improve clarity.
The trinomial expansion can be calculated by applying the binomial expansion twice, setting = +, which leads to (+ +) = (+) = = = = (+) = = = ().Above, the resulting (+) in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index .
Vieta's formulas can be proved by considering the equality + + + + = () (which is true since ,, …, are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of between the two members of the equation.
In the United Kingdom and many other English-speaking countries, "brackets" means (), known in the US as "parentheses" (singular "parenthesis"). That said, the specific terms "parentheses" and "square brackets" are generally understood everywhere and may be used to avoid ambiguity. The symbol of grouping knows as "braces" has two major uses.
Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...
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